Assumptions

Ignore spatial information of the system; we ignore molecular dynamics of the system.

Keep track of number of molecules of each type - concentrations of these state variables.

Thus assume that the system is homogeneous - well-stirred, so that the molecules of each type are spread uniformly throughout spatial domain. *assume thermal equilibrium

assume constant volume of spatial domain

Our models

Introduction

We can condition the system in various manners, but for the purposes of our project, Infector Detector, we will seek a formulation which is valid for both constructs considered.

Our initial approach assumed that energy would in unlimited supply, and that our system would eventually reach steady-state (Model 1). Experimentation suggested otherwise; our system needed to be amended. This lead to the development of model 2, an energy-dependent network, where the dependence on energy assumed Hill-like dynamics:

Model 2: Equations developed through steady-state analysis; however due to limited energy supply, we operate in the transient regime

where:

[A] represents the concentration of AHL-LuxR complex
[P] represents the concentration of pLux promoters
[AP] represents the concentration of A-Promoter complex
k1, k2, k3, k4, k5, k6 are the rate constants associated with the relevant forward and backward reactions represents the energy consumption due to gene transcription. It is a function of gene length.
n is the positive co-operativity coefficient (Hill-coefficient) the half-saturation coefficient

The system of equations for the two constructs varies strictly with respect to the value of the parameter k1. Construct 1 possesses a non-zero k1 rate constant, whereas for construct 2, a zero value is assumed.